The smooth ergodic theory extended to infinite dimensional dynamical systems.
The multiplicative ergodic theorem [1]}, which plays a fundamental role in the smooth ergodic theory and nonuniform hyperbolic theory (see [2]} for more details of this theory) in finite-dimensional dynamical systems, be extended to the infinite setting.
Ruelle [3]} first proved the multiplicative ergodic theorem for compact linear operators on a separable Hilbert space.
Mané [4]} overcame the lack of inner product structure and extended the multiplicative ergodic theorem to compact operators on Banach spaces.
Later, Thieullen [5]} obtained the multiplicative ergodic theorem for continuous bounded linear operator cocycles on Banach spaces.
Lian and Lu [6]} established the multiplicative ergodic theorem for strong measurable operator cocycles on separable Banach spaces.
Froyland [7]} extended it to a continuous semi-invertible cocycle on Banach spaces, and Tokman et al [8]} set up the multiplicative ergodic theorem for a strong measurable case on Banach spaces.
The nonuniform hyperbolic theory in infinite dimensional dynamical systems also be studied,
Lian, Lu [6]} and Varzaneh, Riedel [10]} construct the unstable and stable manifolds on random dynamical systems on a certain Banach space and a fields of Banach spaces.
The more results about infinite dimensional dynamical systems can refer to [11]}, [12]}, [13]}, [14]}, [15]}, [16]}.
